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Let $U$ be a punctured disk, and let $G\to U$ be a quasi-finite separated group scheme. (Assume $K$ of char zero if it helps)

Why is $G = G_1\sqcup G_2$, where $G_1 \to U$ is finite and $G_2\to U$ lies over Spec $K$?

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  • $\begingroup$ The last question has a negative answer, because you don't have any hypotheses that distinguish $0$ from any other point in the disk. You may want to specify that the pullback of $G$ to the punctured disk is étale, or something. $\endgroup$ – S. Carnahan Dec 11 '14 at 11:51
  • $\begingroup$ Yes, you are right. I am assuming that $G$ is in fact generically etale. $\endgroup$ – Maksim Symirno Dec 11 '14 at 13:00
  • $\begingroup$ @KestutisCesnavicius Thank you for your comment. If you post it as an answer, I can accept it. $\endgroup$ – Maksim Symirno Dec 12 '14 at 12:03
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[At OP's suggestion, I am moving my comment to an answer. In the original question $U$ was the spectrum of a Henselian DVR $R$ with $K = \mathrm{Frac}(R)$.]

Define $G_1$ to be the union of those connected components of $G$ whose images cover $U$. Then use [EGA $\text{IV}_4$, 18.5.11 c)] to verify that $G_1$ is a clopen $U$-finite subgroup scheme of $G$.

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